Order

A set $S$ is said to possess an order (or, if necessary to disambiguate, a total order) if, for every pair of distinct elements $x,y\in S$, either $x$ is before $y$, or $x$ is after $y$, with respect to some order relation.

Besides what Objective Mathematics means by "order," there are two other significant uses of the word. One refers to a command (an order) given to other people, as in "he ordered a pizza." In the other usage, a set of things is said to possess an order if there is some pattern to which the things adhere. Both of these usages are related to the mathematical usage of the term; it's not a coincidence that they are all described by the same word. But getting into that would take us too far afield.

Before and after

The concepts of "before" and "after," in their primary usage, are irreducible primaries, and thus admit only ostensive definitions. Nevertheless, I will give a circular "definition" to indicate what is meant by them. The primary usage of these concepts, and the way that they are first arrived at by a conceptual consciousness, is as measurements of relative time-ordering. That is, we directly perceive that some events occur before other events, and that some events occur after other events. Then, by a process of measurement omission, we create the corresponding concepts.

Besides the primary usage, there are man derivative usages of "before" and "after." Consider the following example, wherein paragraphs are sorted from least abstract to most abstract.

1. In most contexts, the number 1 said to be before the number 2, because, when counting, the sound "one" is uttered before the sound "two." Here, the latter usage of "before" is primary; the former is derivative. Likewise, in most contexts, the letter X is said to be before the letter Y, because "X" is always pronounced before "Y" in the alphabet song.
2. Having established an ordering of the letters of the alphabet, we can extend it to an ordering of the English language, by using the rules of alphabetical ordering. Alphabetical ordering tells us, for any two distinct words, which one is before and which is after.
3. We can use alphabetical ordering to give an ordering to many other things. For example, a set of people (say, on an attendance sheet in a school) could be ordered with respect to the alphabetical ordering of their names. In such a situation, Bob may be said to be before Cindy.

What is happening here is that we are using "before" and "after" to define an order-relation on things like the alphabet, the English language, and a set of people.

Order relations

There are some relations which are like "before" and "after" in some respect, but mean something else. The concept subsuming such relations is that of an order-relation.

Examples

"To the right of" and "to the left of"

"On top of" and "underneath"

"Higher pitch than" and "lower pitch than"

"North of" and "south of"

"Later in the alphabet than" and "earlier in the alphabet than"

The traditional concept

Standard mathematics says that, by definition, an order relation on a set $S$ is a function $R:S\times S\rightarrow \{0,1\}$ such that, for any $x,y,z\in S$, it satisfies

1. Reflexivity: $R(x,x)=1$
2. Antisymmetry: If $R(x,y)=1$ and $R(y,x)=1$, then $x=y$.
3. Transitivity: If $R(x,y)=1$ and $R(y,z)=1$, then $R(x,z)=1$.

Although I accept that any order relation will indeed satisfy properties (1) through (3), I do not think I accept the above as a definition of an order relation. I think it is inconsistent with the principles of definition laid out in Concepts#Definition. For the time being, however, I don't have a better definition.

Partial order

A partial order is like a total order, but not every pair of elements of the set are required to be before or after others. [TODO define properly?].

Examples

The leaves of a tree (a real, physical tree; not a mathematical tree) possess a partial order with respect to their height above the ground. For some ordered pairs of leaves, the first is higher than the second. For others, the first is lower than the second. For yet others, it cannot be said that the first leaf is higher than the second, nor can it be said that the first leaf is lower than the second; they are at approximately the same height.